We study the d-dimensional vector bin packing problem, a well-studied generalization of bin packing arising in resource allocation and scheduling problems. Here we are given a set of d-dimensional vectors v1, . . . , vn in [0,1] d , and the goal is to pack them into the least number of bins so that for each bin B, the sum of the vectors in it is at most 1 in every dimension, i.e., || v i ∈B vi||∞ ≤ 1. For the 2-dimensional case we give an asymptotic approximation guarantee of 1 + ln(1.5) + ǫ ≈ (1.405 + ǫ), improving upon the previous bound of 1 + ln 2 + ǫ ≈ (1.693 + ǫ). We also give an almost tight (1.5 + ǫ) absolute approximation guarantee, improving upon the previous bound of 2 [23]. For the d-dimensional case, we get a 1.5 + ln() + ǫ ≈ 0.807 + ln(d + 1) + ǫ guarantee, improving upon the previous (1+ln d+ǫ) guarantee [2]. Here (1 +ln d) was a natural barrier as rounding-based algorithms can not achieve better than d approximation. We get around this by exploiting various structural properties of (near)-optimal packings, and using multi-objective multi-budget matching based techniques and expanding the Round & Approx framework to go beyond rounding-based algorithms. Along the way we also prove several results that could be of independent interest.
We study the two-dimensional bin packing problem with and without rotations. Here we are given a set of two-dimensional rectangular items I and the goal is to pack these into a minimum number of unit square bins. We consider the orthogonal packing case where the edges of the items must be aligned parallel to the edges of the bin. Our main result is a 1.405-approximation for two-dimensional bin packing with and without rotation, which improves upon a recent 1.5 approximation due to Jansen and Prädel. We also show that a wide class of rounding based algorithms cannot improve upon the factor of 1.5.
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